Quantum error correction with realistic measurement data

ABSTRACT

A quantum computing system computes soft information quantifying an effect of soft noise on multiple rounds of a syndrome measurement that is output by a quantum measurement circuit. The soft noise arises due to imperfections in a readout device that introduce variability in repeated measurements of ancilla qubits and is distinct from quantum noise arising from bit-flips in data qubits that are indirectly measured by the ancilla qubits. The quantum computing system applying decoding logic to identify fault locations within the quantum measurement circuit based on the computed soft information.

BACKGROUND

The scalability of decoders for quantum error correction is an ongoingchallenge in generating practical quantum computing devices. Hundreds orthousands of high-quality qubits with a very low error rate (e.g., 10⁻¹⁰or lower) may be needed to implement quantum algorithms with industrialapplications. Due to the high noise rate of quantum hardware, extensivequantum error corrections necessary in the design of a large scalequantum computer. One of the most popular quantum error correctionschemes for fault-tolerant computing is the surface code because it iseasy to implement on a grid of qubits using only logical gates andbecause it tolerates high error rates. Numerical simulations show thatthe surface code achieves a good performance for a variety of noisemodels, even when implemented with noisy quantum gates or in thepresence of coherent errors. However, the noise model for the surfacecode is generally very simplistic. While schemes have been devised toaccount for quantum noise (e.g., bit flips or phase flips on dataqubits), existing noise models do not propose any solution that accountsfor analog classical noise on the measurement outcome that arises due toimperfections in the readout device.

SUMMARY

According to one implementation, a system includes a soft informationcomputation engine computes soft information quantifying an effect ofsoft noise on multiple rounds of a syndrome measurement output by aquantum measurement circuit. The soft noise is noise arising fromimperfections in a readout device and/or limited measurement timeintroducing variability in repeated measurements of ancilla qubits. Adecoding unit uses the computed soft information to identify faultlocations that collectively explain the measured syndrome

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter. Otherimplementations are also described and recited herein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example quantum computing system that utilizessoft outcomes from quantum measurements to perform error detection andcorrection.

FIG. 2A illustrates an example portion of a syndrome measurement circuitthat employs surface code to provide syndrome data to a decoding unit ina quantum computing device.

FIG. 2B illustrates a graph structure that is representative of ameasured syndrome using the syndrome measurement circuit of FIG. 2A.

FIG. 3 illustrates an example of decoding logic implemented by adecoding unit.

FIG. 4 illustrates an exemplary decoding logic that utilizes softoutcome data to compute soft vertical edge weights for a decoding graph.

FIG. 5 illustrates example decoding logic that relies on edge weightscomputed based on soft information.

FIG. 6 illustrates example operations computed by different componentsof a exemplary quantum computing system that uses soft information toimprove performance of a decoding unit.

FIG. 7 illustrates exemplary computing environment suitable forimplementing aspects of the disclosed technology.

DETAILED DESCRIPTION

The herein disclosed technology provides a noise model and decoder thatuses the noise model to correct errors detected in a qubit register of aquantum computer. Unlike other decoding solutions, the disclosed designimplements corrections that account for both quantum noise and classicalnoise, yielding a higher error correction capability and betterperformance than currently-competing decoders. In quantum computing,“quantum noise” refers to noise that can flip the state of the system,resulting in a flip of the outcome that is measured by a readout device.Classical noise (also referred to herein as “soft noise”), in contrast,refers to noise that arises due to imperfections in the readout deviceand/or limited time for sampling the measurement from the readoutdevice. In a quantum device, soft noise introduces variability inrepeated measurements of ancilla qubits.

This noise arises in the measurement of the quantum outcome rather thanin the underlying state of qubits captured by the measurement and is, inmost all quantum systems, widely ignored.

Flash memory devices provide an on-point example of how classical noisecan impact a readout device. In flash memory, each flash cell isprogrammed to store a voltage state that may be directly interpreted asrepresenting one or more bits of data. For example, in a multi-levelcell in flash memory may be programmed to stored one of four voltagestates, each representing one of the binary states: 11, 10, 01, and 00.Due to imperfections in the readout hardware of flash memory devices,actual voltages stored in each flash cell may vary slightly around thetargeted value. When, for example, a flash cell read operation returns avoltage that is somewhere between the voltage representing two distinctstates (e.g., binary state ‘11’ and ‘01’), the device must implementlogic to select the binary state that most accurately corresponds to thestored volage. Flash memory devices typically employ classical errorcorrection code to implement this logic.

Like flash memory, imperfections of a quantum read-out device can alsoimpact the interpretation of the measured outcome. However, there existsa stark difference between the classical (flash memory) and quantumapplications that severely complicate and limit the efficacy ofclassical error correction at the quantum level. These differences arisefrom the fact that flash memory permits a direct measurement of theencoded data within each cell. In contrast, data qubits cannot bedirectly measured in quantum applications without destroying theunderlying data. For this reason, quantum readout devices performindirect measurements (e.g., typically, by entangling a group of dataqubits with an ancilla qubit and measuring the ancilla qubit to inferthe parity of the entangled data qubits). As a consequence of the above,existing classical corrections that account for read-out device error donot work for quantum applications.

The herein disclosed systems provide quantum device corrections thataccount for classical Gaussian noise in a readout device that shifts theoutcome of a measurement by a random variable with normal distribution.According to one implementation, the disclosed decoding solutionsreceive and interpret quantum measurements using “soft” outcome datarather than the traditionally-used “hard” outcome data. As used herein,the term “hard outcome” refers to a binary value (e.g., 1/0) whereas a“soft outcome” refers to a real (non-binary) number. According to oneimplementation of the herein disclosed technology, a quantum readoutdevice measures soft outcomes, where each soft outcome is a real numbermeasurement ‘o’ sampled from any pair of probability distributionsdescribing classical noise, where one of the distributions is associatedwith the outcome +1 and the other is associated with the outcome -1. Inthe example of FIG. 1 (discussed below), the outcomes are shown to beGaussian but these outcomes can also be non-Gaussian distributions.Traditional quantum devices discard this soft value by immediatelymapping soft outcomes to hard (binary) outcomes. For example, -1.2 maybe mapped to ‘-1’ while +0.8 may be mapped to 1, where -1/1 are thenrepresented as binary digits. This type of mapping discards data thatmay have meaningful application in error correction. The followingmethodologies propose novel models that allow for this soft outcome datato be used in quantum error correct, ultimately improving detectorperformance.

FIG. 1 illustrates an example quantum computing system 100 that utilizessoft outcomes from quantum measurements to perform error detection andcorrection. The system includes a decoding unit 116 that includes asub-engine referred to herein as a soft information computation engine122. The soft information computation engine 122 uses soft measurementoutcomes (discussed below) to compute various metrics that are providedas an input to a primary decoder 120. The primary decoder 120 mayimplement logic typical of any standard decoder whose role is toidentify the error affecting qubits and correct the resulting error. Byexample and without limitation, the primary decoder 120 may implementlogic utilized by decoders such as the Minimum Weight Perfect Matching(MWPW) Decoder, the Union-Find (UF) Decoder, as well as other machinelearning decoders or tensor network decoders. Notably, the primarydecoder 120 differs from traditional implementations in that it acceptssoft inputs (e.g., metrics computed by the soft information computationengine 122) in lieu of hard inputs that are used in traditionalapplications. This yields increased decoding performance.

The quantum computing system 100 includes a controller 102 that performscalculations by manipulating qubits within a qubit register 108. Toenable fault tolerant quantum computation in the quantum computingsystem 100, a readout device 112 includes a syndrome measurement circuit114 that applies quantum error correction code(s) (QECCs) to data readout from the qubit register 108 by the readout device 112. QECCs havebeen developed to protect quantum states from quantum noise duringmeasurement. In any QECC, logical qubits are encoded using severalphysical qubits to enable fault tolerant quantum computations

Since measurement is known to destroy the delicate states of a qubitneeded for computation, the syndrome measurement circuit 114 usesredundant qubits—known as “ancilla data bits” to perform computations.During quantum processing, entropy from the data qubits that encode theprotected data is transferred to the ancilla qubits (sometimes alsoreferred to as “measurement qubits”) that can be discarded. The ancillaqubits are positioned to interact with data qubits such that it ispossible to detect errors by measuring the ancilla qubits and to correctsuch errors using a decoding unit 116 that includes one or moredecoders. In some implementations, the decoding unit 116 includes logicexecuted by one or more classical computing systems.

The syndrome measurement circuit 114 performs measurements of theancilla bits in the quantum computer to extract syndromes providinginformation measured with respect to quantum errors (faults). In orderto avoid accumulation of errors during the quantum computation, thesyndrome data is constantly measured, producing r syndrome bits for eachsyndrome measurement round. By example and without limitation, view 118provides a visualization of three rounds of syndrome measurement data(e.g., at times t=0, t=1, and t=2). At each round, a measurement isperformed on a number of ancilla qubits represented as open circles(e.g., an ancilla qubit 126) on a 2D grid. Each of the ancilla qubitsencodes the parity state of a set of group of entangled neighboring dataqubits. This concept is roughly illustrated by view 128, illustratingthe ancilla qubit 126 that encodes a parity state of four neighboringdata qubits (shown as filled black circles). Although each of theancilla qubits may be similarly understood as encoding parity data ofneighboring data qubits, the data qubits are omitted from the remainderof view 118 for simplicity. Thus, the open circles all represent ancillaqubits.

At each of rounds t-0, t=1,and t=2, the ancilla qubits are measured andthe resulting data is read out by the readout device 112 as an arrayreferred to herein as a “syndrome” that includes outcome values for eachround of measurement. Notably, the data bits of the syndrome are readout and provided to the decoding unit 116 as soft outcomes (real numbervalues).

When the state of an ancilla qubit (e.g., ancilla qubit 126) isphysically measured, the measurement produces one of two different basisstates |0

and |1

. This information may be conveyed in different forms, depending on thespecific quantum architecture used to implement the individual qubits.In one implementation, the readout device 112 is designed to generateoutcome data corresponding to binary values or other integer valueseasily mapped to binary, such as -1 and 1. Due to imperfections in thereadout device 112 and limits on measurement time for sampling eachround of syndrome measurement, the actual values returned by the readoutdevice 112 have some natural probabilistic variation centered aroundthese target outcomes, as illustrated by view 130.

Specifically, view 130 shows a first distribution 132 of soft outcomescentered at integer value -1 and a second distribution 134 of softoutcomes centered at integer value 1. Stated differently, actual valuesreturned by the readout device 112 are “soft values” that fall withindistributions centered at each integer value (-1, 1), corresponding tohard outcomes of 0 and 1, respectively. For example, a real outcome(“soft outcome”) may be -1.2 rather than -1, as shown. The twodistributions 132, 134 may be understood as having integrals of equalvalue and as being Gaussian distributions (as shown) or distributions ofany family of classical distribution (e.g., any non-gaussian type ofclassical distribution) .The first distribution 132 and the seconddistribution 134 can be the same or different from one another and/orhave different parameters. For example, one distribution can be Gaussianand one Poisson, both Gaussian with different variance σ and σ′, etc.The two distributions representing the range of soft outcomes may varywith time and may depend on the state of the qubit being measured.

In traditional quantum applications, soft outcome values are immediatelymapped by the decoding unit 116 to a hard outcome value using a naiveapproximation: ô=sign(o), where o is the soft outcome value. Per thismapping the soft outcome data is discarded and analysis of the decodingunit 116 hinges instead on the hard outcome data. For example, the softoutcome -1.2 is mapped to the hard outcome -1, and the soft outcome +0.4is mapped to the hard outcome value of 1.0. In the presently disclosedtechnology, however, the syndrome at each measurement round is providedin the form of a soft outcome vector that includes the soft measurementfor each ancilla qubit in the readout device 112. This is repeated overmultiple measurement rounds, as shown.

The multiple rounds of syndrome data are provided to the softinformation computation engine 122, which computes further metrics usingthe soft data including “soft edge weights,” which are discussed ingreater detail below with respect to FIGS. 4-6 . According to oneimplementation, the soft information computation engine 122 applies anovel graphical model for “soft noise” with continuous outcomes. Theterm “soft noise” is used interchangeably herein with “classical noise”to refer to noise that arises due to imperfections in the readout deviceand/or limits on measurement time for performing the measurements of themeasurement device. In a quantum device, soft noise introducesvariability in repeated measurements of ancilla qubits and is distinctfrom quantum noise arising from bit-flips in data qubits that areindirectly measured by the ancilla qubits.

Soft noise metrics based on this noise model may be used by the primarydecoder 120 in lieu of or in addition to traditionally-used “hardoutcome” inputs to improve detector performance. The primary decoder 120outputs error solution information that identifies fault locations(e.g., locations of quantum bit-flips) within the readout device 112that allows the controller 102 to perform error correction.

FIG. 2A illustrates example portions of a syndrome measurement circuit200 that employs surface code to provide syndrome data to a decodingunit in a quantum computing device. The syndrome measurement circuit 200may be integrated within a quantum computing system to provide softoutcome syndrome data to a primary decoder, such as decoder 120 asdiscussed with respect to FIG. 1 .

In the surface code, physical qubits are entangled together using asequence of physical qubit CNOT operations, with subsequent measurementsof the entangled states providing a means for error correction and errordetection. A set of physical qubits entangled in this way is used todefine a logical qubit, which due to the entanglement and measurementhas far better performance than the underlying physical qubits.

In FIG. 2A, the syndrome measurement circuit 200 encodes a singlelogical qubit onto a square grid 202 of dxd data qubits, where d isreferred to as the minimum distance of the surface code. In the exampleof FIG. 2A, the grid 202 represents a two-dimensional distance-3 surfacecode. In the simplest realization, a surface code of distance d uses(2d-1)² qubits to store a single logical qubit, where d is a measure ofredundancy and error tolerance. A larger code distance results ingreater redundancy and increased error tolerance. Thus, the square grid202 shown may be understood as representing a single logical qubit thatis one of many different logical qubits measured by the syndromemeasurement circuit 200 in the quantum computing system.

In the particular representation shown in FIG. 2A, data qubits are shownby hollow circles on vertices of the grid (e.g., a data qubit 204).Although not shown, the face of each plaquette (e.g., a square plaquette208) may be understood as including an ancilla qubit that is coupled toeach of the data qubits at the four corners of the plaquette. Errorcorrection is based on the measurement of r=d²-1 syndrome bits that areextracted from a qubit register by the syndrome measurement circuit 200.

In the syndrome measurement circuit 200 of FIG. 2A, data qubit errors(e.g., quantum noise) can be modeled by introducing random X bit-flipand Z phase-flip operations in the evolution of the qubit state where Xand Z are the Pauli X-gate and the Pauli Z-gate, respectively. Note, thePauli-Y gate is the combination Ŷ={circumflex over (X)}{circumflex over(Z)}, and is thus usable to model a combined bit flip error and phaseflip error. The operator model for correcting single-qubit errorsimplies that these errors can, in principle, by undone by applying the Xand Z correction gates (e.g., an erroneous Z-type error can be canceledby subsequently applying an intentional {circumflex over (Z)}, since{circumflex over (Z)}²=Î (the identity matrix).

The syndrome measurement circuit 200 utilizes two types of ancillaqubits referred to herein respectively as “measure-Z” qubits (centeredon plaquettes with shaded faces, such as shaded plaquette 208) and“measure-X” qubits (centered on plaquettes with unshaded faces, such asunshaded plaquette 210). In surface code literature, the measure-Z andmeasure-X qubits are sometimes called Z syndrome and X syndrome qubits.In the surface code configuration of FIG. 2 , each measure-X andmeasure-Z qubit is coupled to four data qubits (e.g., the four nearesthollow circles). Further, each data qubit is coupled to two measure-Zqubits and two measure-X qubits (e.g., the four nearest solid filledcircles). A measure-Z qubit is used to force is used to force itsneighboring data qubits a, b, c, and d into an eigenstate of theoperator product {circumflex over (Z)}_(a), {circumflex over (Z)}_(b),{circumflex over (Z)}_(c), {circumflex over (Z)}_(d). Likewise, ameasure-X qubit is used to force is used to force its neighboring dataqubits a, b, c, and d into an eigenstate of the operator product{circumflex over (X)}_(a), {circumflex over (X)}_(b), {circumflex over(X)}_(c), {circumflex over (X)}_(d). Each measure-Z qubit thereforemeasures what is known as a Z-stabilizer and each measure-X qubitmeasures what is known as an X-stabilizer. Stabilizers (e.g., theX-stabilizer and the Z-stabilizer) are very important in preservingquantum states. By repeatedly measuring a quantum system using acomplete set of commuting stabilizers, the syndrome measurement circuit200 forces the data bits associated with each measure-X and measure-Zqubit into a simultaneous and unique eigenstate of all of thestabilizers allowing one to measure the stabilizers without perturbingthe system. When the measurement outcomes change, this corresponds toone or more qubit errors and the quantum state is projected by themeasurements onto a different stabilizer eigenstate.

The measurement of each ancilla qubit (the measure-Z and measure-Xqubits) is performed by implementing a sequence of operations in a veryparticular order that is effective to entangle the ancilla qubit on theface of each plaquette with its four nearest-neighbor data qubits. Thelast step in this sequence of operations is a measurement of the ancillaqubit (measure-Z or measure-X) in the center of the plaquette. Themeasurement of this ancilla qubit returns a syndrome bit correspondingto the parity of the number of errors on the qubits of the plaquette.For example, a 0-value bit indicates a trivial syndrome (implying eitheran absence of fault or an even number of faults that cancel out) while a1-value bit indicates a non-trivial syndrome (either a single fault oran odd number of faults).

FIG. 2B illustrates a graph structure 212 that is representative of ameasured syndrome using the syndrome measurement circuit of FIG. 2A.This type of structure is referred to herein as a “decoding graph.” Thegraph structure 200 represents a 2D decoding graph of the form graphG=(V, E) for the correction of X-type errors is a graph with vertex setV=V_(⋅)∪ V_(●), such that each edge of G corresponds to a qubit of thesurface code and each node of V_(⋅)corresponds to a Z measurement. Thenodes of V_(⋅)are referred to herein as the measurement nodes, while thenodes of V_(●)are herein referred to as boundary nodes. In FIG. 2B,measurement nodes are black and boundary nodes are white.

The 2d decoding graph of FIG. 2B provides a convenient representation oferrors and syndromes. A X error pattern is represented by a binary errorvector x ϵ

such that x(e)=1 if and only if the edge e suffers form a X error. Givenan error, if it is assumed that the measurement circuitry is ideal, around of measurement returns the outcome vector m ϵ

such that the measurement outcome in node v is

$\begin{matrix}{{\overset{\_}{m}\left( {v,x} \right)} = {\left( {\underset{v \in e}{\sum\limits_{e \in E}}{x(e)}} \right){mod}2}} & (1)\end{matrix}$measuring the parity of the number of errors incident to v.

Quantum decoders typically extend the above model to account for qubiterrors introduced during measurement. This is done by performing errorcorrection based on multiple consecutive rounds of measurement data(e.g., as shown in FIG. 1 with respect to the syndrome re-measured att=0, t=1, t=2, etc.). One common model accounting for errors introducedduring measurement is the phenomenological noise model, which describesconsecutive rounds of syndrome extraction as a sequence N₁, M₁, N₂, M₂,. . . alternating noise cycles N_(t) and noisy measurement rounds M_(t).A noise configuration for T rounds of syndrome extraction withphenomenological noise is a sequence x₁, ƒ₁, . . . , x_(T),ƒ_(T) wherex_(t) ϵ

is the error occurring during the noise cycle N_(t) and ƒ_(t) ϵ

represents outcome flips during the t th measurement round. Moreprecisely, x_(t)(e)=1 if an error occurs on qubit e during N_(t) andƒ_(t)(v)=1 if and only if the outcome of the measurement of v duringM_(t) is flipped. Based on Eq.(1), the syndrome extraction round M_(t)returns the outcome vector m _(t) ϵ

such that the outcome in node v ism _(t)(v)=(m(v, x)+ƒ_(t)(v))mod2   (2)where x=(x₁+ . . . +x_(t))mod2. The outcome vector m _(t) depends on xwhich is the accumulation of all errors occurring before thismeasurement. Existing decoders generally approximate this outcome bymapping soft outcomes to hard outcomes. Exemplary hard outcome syndromebit values (1s and 0s) are shown at the measurement nodes in FIG. 2B. Asis discussed further below, the herein disclosed techniques provide forutilizing the soft outcome data, at least in part, to calculateparameters used by a decoding unit.

FIG. 3 illustrates an example of decoding graph 300 constructed by adecoding unit employing logic to identify fault locations affecting asyndrome measurement. For simplicity, the following discussion isgeneralized to logic that may be implemented by a variety of differenttypes of decoding units. As explained with respect to FIG. 1 , thedecoding unit receives multiple rounds t of a syndrome, where each roundof the syndrome includes an array of values.

In general, the objective of the decoding logic illustrated by thedecoding graph 300 is to identify a set of fault paths that explain thereceived syndrome S for X-type or Z-type measurements received from thesyndrome measurement circuit. It has been mathematically proven that themost likely fault configuration is given by what is referred to hereinas “the minimum weight solution.” Thus, the decoding unit employs logicto derive or estimate a minimum weight solution for a predefined numberof syndrome measurement rounds.

As used herein, the term “minimum weight solution” refers to a set ofpaths formed along edges of the decoding graph that (1) collectivelyexplain the observed non-trivial syndrome bits for a predefined numberof measurement rounds where (2) the total summed weights of theindividual paths in the set is a minimum among the set of pathssatisfying (1). The “weight” or “distance” of a path is closely relatedto (and sometimes equated with) the number of edges in a path orsolution set. For example, some decoders may employ logic providing thata solution consisting of one single edge has a weight of 1, while otherdecoders employ slightly different definitions. Exemplary definitionsfor edge weight are set forth herein with respect to FIG. 4 .

The term “minimum weight path” is herein defined as a singular pathformed by contiguous edges in the decoding graph, where the combinedweight of all edges within the path is of a minimum total weight of anyvalid path that exists to explain non-trivial bit values correspondingto endpoints of the path. It is known that, statistically, a minimumweight path is very close to being the most likely fault path betweentwo non-trivial syndrome bits.

In the illustrated example, the decoding graph 300 includes fournon-trivial nodes (filled circles) that are observed over threemeasurement rounds. A first path 302 of length 1 (one edge) is shownbetween non-trivial syndrome nodes 304, 306. A fault that occurs alongthe path 302 fully explains the non-trivial syndrome nodes 304, 306.Thus, the first path 302 is a minimum weight path that is one ofmultiple paths in the illustrated minimum weight solution. A second path308 extends between non-trivial nodes 310, 312. This path has a lengthof 2-edges, where a fault on each of the two edges would explain thenon-trivial nodes 310, 312 (e.g., the even-number of faults affectingtrivial node 314 causes this node to be trivial rather thannon-trivial). This is also a minimum weight fault path. Collectively,the paths 302 and 308 represent the minimum weight solution to thedecoding graph 300.

According to one implementation, the decoding unit computes the minimumweight solution for the decoding graph 300 by computing all possiblepaths that collectively explain the faults, computing the total “weight”of each path (net sum of edge weights), and then selecting the solutionset that is of minimum total weight. Per the novel disclosed methodology(discussed further below), the weight of each of the edges in thedecoding graph is computed based on soft outcome data. This improvesaccuracy of the decoding algorithm as compared to when the decodingmethodology calculates edge weights based exclusively on hard outcomedata.

FIG. 4 illustrates an exemplary decoding logic 400 that utilizes softoutcome data to compute soft vertical edge weights for a soft decodinggraph 410. Here, soft edge weights are derived using a soft noise model,discussed below. Existing noise models such as the phenomenologicalnoise model do not provide any mechanism or metric to account for thesoft noise that arises due to imperfections in the quantum readoutdevice and/or time constraints imposed on measurement rounds of thequantum readout device. Due to this soft noise, the data extracted froma syndrome measurement circuit is not a binary value but instead a realnumber.

In practice, the value m _(t)(v) (in Eqn. (2) above) cannot be measured.However, a naive estimation of this value can be attained by mappingeach soft outcome value to a hard outcome value, {circumflex over (m)} ϵ

defined by:

$\begin{matrix}{{{\hat{m}}_{t}(v)} = \left\{ {\begin{matrix}0 & {{{{if}{m_{t}(v)}} \geq 0},} \\1 & {{{if}{m_{t}(v)}} < 0}\end{matrix}.} \right.} & (3)\end{matrix}$

An exemplary plot 402 in FIG. 1 illustrates this concept. In the plot402, curves 404, 406 are assumed to have equivalent integral values. Thecurves may be gaussian or non-gaussian, as given by any probabilitydistribution. Here, soft outcomes greater than 0 are mapped to a hardoutcome of zero while soft outcomes less than 0 are mapped to a hardoutcome of 1. Using the above relationship in Eqn. (3) to map softoutcome data to hard outcome data reduces performance of the detectorbecause information is inevitably lost, at times, when the soft outcomeis mapped to the wrong hard outcome value .

Rather than use the naive approximation of Eqn. (3) to map soft outcomedata to hard outcome values and thereby discard the soft outcome values,the below-proposed soft noise model may provide a methodology forutilizing measured soft syndrome values in a way that more accuratelyaccounts for the soft noise arising from within the measurementapparatus.

Soft noise model

Consider a sequence of T rounds of measurement for a surface code withqubit noise , x₁, . . . , x_(T) and outcome flips ƒ₁, . . . , ƒ_(T). Thet-th round of measurement returns a soft outcome vector m_(t) ϵ

defined bym _(t)(v)=(-1) ^(m) ^(t) ^((v))+A _(t,v)   (4)where m _(t)(v) is defined in Eq. (2) and A_(t,v) is the “soft noise”that can be any family of distribution that contains one distributionA_(t,v) for each ancilla qubit and for each time step t. As an exxample,we can consider the normal distribution A_(t,v)˜N(0, σ) with variance σϵ

Graphical Model For Soft Noise Distribution

In this section we provide a graphical representation of the errorsdistribution. It is an extension of the standard decoding graph (e.g.,as shown in FIG. 3 ), that is based on a consideration of softmeasurement data. An example soft decoding graph 410 is shown in FIG. 4as corresponding to T=3 rounds of measurements for the surface code.There is a copy of the 2d decoding graph for each time step and anadditional copy supporting no measurement is added for a propertermination.

Each measurement node v_(t) (hollow and filled circles) at time t isconnected to its counterpart v_(t+1) at the next time step. Forsimplicity, the soft decoding graph 410 represents a single verticaledge connecting v_(t) and v_(t+1) but this graph is better understood asincluding a double edge between each vertical pair of nodes-a hard edgeand a soft edge. To better illustrate this concept, an edge (B) from thedecoding graph 410 is shown in expanded view 412 as including twocounterparts—edges B1 and B2, wherein B1 is a hard edge (straight) andB2 is a soft edge (curved) extending in the time dimension between twomeasurements of a same node. In the soft decoding graph 410, horizontaledges correspond to qubit errors and vertical edges representing thetime dimension (e.g., hard edge B1 and soft edge B2) encode hard andsoft outcome flips, respectively.

We say that a hard flip occurs in node u if the hard outcome switchesvalue between two consecutive time steps (e.g.,. v_(t) and v_(t+1)).

We say that a soft flip occurs in node v between steps t and t+1 if thehard outcome (defined via equation (3)) does not equal the outcomevector (defined by equation (2) (e.g., {circumflex over (m)}_(t)(v)≠m_(t)(v)).

Consider an error configuration for T rounds of measurement with qubiterrors x₁, . . . , x_(T) with hard flips ƒ₁, . . . , ƒ_(T) and softflips , {tilde over (ƒ)}₁, . . . , {tilde over (ƒ)}_(T). The softdecoding graph is built in such a way that each edge corresponds to apossible fault in our noise model. An horizontal edge of the t th layerof G corresponds to a qubit error during N_(t). A hard (respectivelysoft) vertical edge between v_(t) and v_(t+1) corresponds to a hard flipƒ_(t)(v) (respectively soft flip {tilde over (ƒ)}_(t)(v)). As a result,we can consider such an error configuration as an error vector ε ϵ

.

Edges of the horizontal edges (corresponding to qubit errors) depend onthe qubit noise rate p_(q) and the measurement outcome flip rate p_(ƒ).During a noise cycle N_(t), each qubit is affected by an error withprobability p and errors are independent. During a measurement roundM_(t), the outcome of the measurement of node v is flipped independentlywith probability p_(ƒ). Given the noise parameters p_(q) and p_(ƒ)andthe sequence of soft outcomes m₁, . . . , m_(T), we define edge weightsfor the soft decoding graph 410.

In view of the foregoing, the weight of horizontal edges is, in oneimplementation, set to:w _(q)=-log(p _(q)/(1-p _(q)))   (5)while the weight of hard vertical edges (e.g.,. weight of B1) is:w _(ƒ)=-log(p _(ƒ)/(1-p _(ƒ)))   (6)and the weight of the soft vertical edges {v_(t), v_(t+1)}, alsoreferred to herein as a the “soft edge weight,” (e.g., weight of B2) isdefined to be:

$\begin{matrix}{{w_{soft}\left( \left\{ {v_{t},v_{t + 1}} \right\} \right)} = {{- \log}\frac{G\left( {o^{\prime},\sigma,{m_{t}(v)}} \right)}{G\left( {o,\sigma,{m_{t}(v)}} \right)}}} & (7)\end{matrix}$where o is the hard outcome o=(-1)^({circumflex over (m)}) ^(t) ^((v))in node v at round t and o′=-o is the flipped hard outcome. Note that bydefinition of the hard outcome {circumflex over (m)}_(t)(v), this edgeweight is non-negative. The soft edge weight given by Eqn. (7)represents a likelihood-ratio test (ratio of probability), and morespecifically, a ratio between the probability that the observed softoutcome value comes from either a -1 or a 1, as generally shown withrespect to the plot 402. Here, the numerator on the right-hand side ofEqn. (7) represents the -log of a probability density function of thesoft outcome centered on the opposite of the hard outcome while thedenominator on the right-hand side represents the -log of a probabilitydensity function of the soft outcome centered on the hard outcome.

In general, the ratio expressed by Eqn. (7) above gets smaller as thesoft outcome for a particular bit approaches the hard outcome value forthe bit. This ratio gets larger as the soft outcome approaches 0 (e.g.,approaches the furthest point possible from the hard outcome value thatstill shares the sign (+/-) of the hard outcome value). What this meansthat when the soft outcome value is near zero, there exists a lowconfidence in the accuracy of the hard outcome value at the giventimestep. In this case, the soft edge weight of the correspondingvertical edge (e.g., B2) approaches 0 (e.g., -log(˜1/1)=0). At the sametime, when the soft outcome value is closer to the hard outcome, thereexists a higher confidence in the accuracy of the hard outcome value. Inthis case, the soft weight of the corresponding vertical edge becomesvery large (e.g., -log(value close to 0)=large value).

In effect, the soft edge weight expression given above in Eqn. (7)relies on soft values to adjust the vertical edge weight to be greaterthan 1 when there is a high confidence in the hard outcome and to adjustthe vertical edge weight to trend toward 0 as the confidence in the hardoutcome decreases. When Eqn. (7) is used to calculate vertical edgeweights that are used by a decoder (e.g., the edge weights used by adecoding unit such as the MWPM decoder or the UF decoder), the soft edgeweights bias the decoder toward selecting solution sets including edgesthat represent low-confidence hard flips as prospective fault locations.

For instance, FIG. 4 illustrates an example whereby edge ‘B’ of thedecoding graph represents a low-confidence hard flip. This is becausethe soft outcome value (e.g., 0.1) of the upper node is very close to 0,which is the midpoint between the two distributions. Via the naivemapping provided by Eqn. (3) (soft outcome to hard outcome), this nodeexperiences a hard flip between t=1 and t=2; however, the confidence inthis flip is very low due to the actual soft outcome value at t=2. Thislow confidence makes it more likely that the upper node (at t=2) on edgeB is actually non-trivial (detecting a nearby fault), meaning that it isappropriate and desirable to include edge B in a minimum weight solutionthat is output by the decoding unit.

By modifying the decoding unit to utilize soft edge weight values forthe vertical edges of the decoding graph 410, the decoder is more likelyto select a more accurate minimum weight solution. For example, at eachsyndrome round (t=0, t=1, t=2, etc.) the soft edge weights are computedfor the round using Eqn. (7) (e.g., by the soft information computationengine of FIG. 122 ) and these weights are provided to the decoding unitfor the vertical in place of traditionally-computed vertical edgeweights.

FIG. 5 illustrates exemplary decoding logic that uses soft informationto compute edge weights for a single measurement round of a decodinggraph 500.

The decoding graph 500 is shown to include two rounds of measurement(t=0 and t=1). The decoding graph 500 may be understood as includingnodes corresponding to hard outcome values. Specifically, open (white)nodes correspond to trivial hard syndrome values and filled (black)nodes correspond to non-trivial hard syndrome values. Although notshown, it is assumed that all horizontal edges have weights computedbased on hard outcome values (e.g., as shown by Eqn. (5) above).Vertical edges, in contrast, have weights computed based on soft outcomevalues (e.g., the real number values actually returned from the syndromemeasurement circuit). For simplicity, the four non-trivial nodes in thedecoding graph 500 are labeled as A, B, C, and D.

To identify fault locations, a decoder seeks to identify a minimumweight solution that would explain non-trivial syndromes observed duringthe previous round of measurement. Although different algorithms mayemploy different logic, one popular approach is to try to identify apath extending between each pair of non-trivial nodes that fullyexplains the non-trivial nodes within the path. Each such path isreferred to herein as a valid candidate path. For example, thenon-trivial nodes A and D can be explained by 1-edge segmentrepresenting a single fault location. This is therefore a validcandidate path that the decoding unit may consider for inclusion in aminimum weight solution. In contrast, a path extending from A to D to Ccan be formed with two edges; however, this path is not a validcandidate path because the existence of a fault along each of the twoedges would not fully explain all non-trivial node values along the path(e.g., the two non-trivial edges amount to an even number that shouldcancel out and cause node D to be trivial rather than non-trivial).

Visually, the identification of valid candidate paths in the decodinggraph 500 can be illustrated by drawing one or more paths between eachpair of non-trivial nodes, where each path intersects, at most, 2non-trivial nodes (e.g., the endpoints). An exemplary non-inclusive setof valid candidate paths is shown in view 502. Notably, it may bepossible to compute multiple valid candidate paths for some pairs ofnon-trivial nodes. This extension is excluded from FIG. 5 for simplicityof illustration.

During the fault identification process for each round of syndromemeasurement, a decoder determines (e.g., computes or receives) edgeweights for each candidate path that is identified as explained above.In one implementation, horizontal edge weights are determineddifferently than vertical edge weights. For example, horizontal edgeweights may be computed utilizing Eqn. (5) above (based on hard outcomevalues) while vertical edge weights are based, at least in part, on softoutcome values. In one implementation, vertical edge weights are “soft”edge weights computed via Eqn. (7) above. In general, when Eqn. (5) isused to compute the horizontal edge weights(w_(q)=-log(p_(q)/(1-p_(q))), horizontal edge weights do not vary muchwith varying values of p. Typical values of p may, for example, rangefrom .00001 to .001, leading to weights between 6.9 and 11.5. Forsimplicity and ease of concept, all horizontal edge weights are shown tobe “1” in FIG. 5 . In contrast to horizontal weights, soft vertical edgeweights computed via Eqn. (7) may exhibit values spanning a larger rangeof values.

After determining edge weights for each of the identified minimum-faultpaths existing within the decoding graph 500, the decoding unit seeks toidentify a minimum weight solution. This solution consists of a discretesubset of the identified paths in which the non-trivial endpoints A, B,C, and D each appear exactly once, where the solution represents theminimum total edge weight of all valid solutions.

In the example of FIG. 5 , the identified valid candidate paths providethree possible solution sets that each consist of two discrete paths.These solutions are shown in view 504. A first of these solutionsincludes the paths AD and BC, and has a total weight of 2.2 (e.g.,1+1.2). A second of these solutions includes the paths AB and CD, andhas a total weight of 3 (e.g., 2+1). A third of these solutions includesthe paths AC and BD, and has a total weight of 4.8 (e.g., 1+1+1+1+0.8),Here, the minimum weight solution is given by the first solution—pathsAD and BC. This solution may, in turn, be output by the decoding unitand provided to an error correction code to classically correct storedvalues corresponding to the qubits affected by the faults identified.

Table 1 (below) provides a mathematical proof demonstrating that one canfind the most likely error configuration in a decoding graph by findinga subset of edges of minimum weight that explain the syndrome (e.g., theminimum weight solution) with vertical edge weights derived using Eqn.(7) above.

TABLE 1 Proposition 1: Any error configuration x = (x₁, . . . x_(T)) forT rounds of measurements with hardflips f = (f₁, . . . , f_(T)) and softflips f = (f₁, . . . , f_(T)) corresponds to a vector ε ∈  

. Moreover, given the soft measurements outcomes m = (m₁, . . . ,m_(T)), we have −log 

(ε|m) = C + Σ_(e∈E) _(T) w(e)ε(e) (8) for some constant C that dependsonly on p_(q), p_(f) and m. We illustrate our result with classicalGaussian noise for simplicity but Proposition 1 holds for any classicalnoise distribution. We only require the random variables A_(v,t) to beindependent. For a general family of random variables A_(v,t), the edgeweight defined in Eq. (7) must be replaced by $\begin{matrix}{{w_{soft}\left( \left\{ {v_{t},v_{t + 1}} \right\} \right)} = {{- \log}\frac{\phi_{v,t}\left( {m_{t}(v)} \middle| {\left( {{{\overset{\hat{}}{m}}_{t}(v)} + 1} \right){mod}2} \right)}{\phi_{v,t}\left( {m_{t}(v)} \middle| {{\overset{\hat{}}{m}}_{t}(v)} \right.}}} & \end{matrix}$ (9) where ϕ_(v,t) is probability density function for theconditional distribution of the soft outcome given the hard outcome innode v at step t. In the discrete case, we replace the probabilitydensity function by the probability distribution. The proof ofProposition 1 (below) is unchanged. Proof. The first part of thisproposition was justified after the definition of the soft decodinggraph. Let us prove that Eq. (7) holds. To compute  

(ε|m), we apply Bayes theorem to obtain  

(ε|m) =  

(x, f|m) ∞  

(m|x, f) 

(x) 

(f) (10) The first equality holds because f is fully determined by x, fand m. Then, we used the independence of x and f to write  

(x, f) =  

(x) 

(f). Therein, we use the abusive notation  

(m|x, f) to represent the probability density function of the continuousrandom variable m conditioned on x and f. We will consider the threeterms of the right hand side of (9) separately. First, we will provethat −log 

(x) = C₁ + Σ_(e∈E) _(T) _(h) w(e)ε(e), (11) for some constant C₁, whereE_(T) ^(h) denotes the set of horizontal edges of G_(T). Given that x issupported on horizontal edges of G_(T), using the independence of qubitnoise we can write${{\mathbb{P}}(x)} = {}{{\underset{{x(e)} = 1}{\prod\limits_{e \in E_{T}^{h}}}{p_{q}{\underset{{x(e)} = 0}{\prod\limits_{e \in E_{T}^{h}}}\left( {1 - p_{q}} \right)}}} = {\left( {1 - p_{q}} \right)^{|E_{T}^{h}|}{\underset{{x(e)} = 1}{\prod\limits_{e \in E_{T}^{h}}}{\left( \frac{p_{q}}{1 - p_{q}} \right).}}}}$By definition of the horizontal edge weight, Eq. (10) follows by takingthe log. Using a similar argumentation for the hard flip, we obtain−log 

(f) = C₂ + Σ_(e∈E) _(T) _(v,hard) w(e)ε(e), (12) for some constant C₂,where E_(T) ^(v,hard) denotes the set of hard vertical edges of G_(T).Finally, we will show that −log 

(m|x,f) = C₃ + Σ_(e∈E) _(T) _(v,soft) w(e)ε(e), (13) for some constantC₃, where E_(T) ^(v,hard) denotes the set of soft vertical edges ofG_(T). By definition of m, we have  

(m|x, f) =  

(m|m). Thanks the independence of the soft noise, we can use Eq. (4)which yields $\begin{matrix}{{{\mathbb{P}}\left( m \middle| \overset{¯}{m} \right)} = {\prod\limits_{v_{t} \in V_{T}}{G\left( {\left( {- 1} \right)^{{\overset{¯}{m}}_{t}(v)},\sigma,{m_{t}(v)}} \right)}}} \\{= {\prod\limits_{\underset{{{\overset{¯}{m}}_{t}(v)} = {{\hat{m}}_{t}(v)}}{v_{t} \in V_{T}}}{{G\left( {\left( {- 1} \right)^{{\overset{\hat{}}{m}}_{t}(v)},\sigma,{m_{t}(v)}} \right)}{\prod\limits_{\underset{{{\overset{¯}{m}}_{t}(v)} \neq {{\hat{m}}_{t}(v)}}{v_{t} \in V_{T}}}{G\left( {\left( {- 1} \right)^{{{\overset{\hat{}}{m}}_{t}(v)} + 1},\sigma,{m_{t}(v)}} \right)}}}}} \\{= {\prod\limits_{v_{t} \in V_{T}}{{G\left( {\left( {- 1} \right)^{{\hat{m}}_{t}(v)},\sigma,{m_{t}(v)}} \right)}{\prod\limits_{\underset{{{\overset{¯}{m}}_{t}(v)} \neq {{\hat{m}}_{t}(v)}}{v_{t} \in V_{T}}}\frac{\left. {G\left( {\left( {- 1} \right)^{{{\overset{\hat{}}{m}}_{t}(v)} + 1},\sigma,{m_{t}(v)}} \right)} \right)}{\left. {G\left( {\left( {- 1} \right)^{{\overset{\hat{}}{m}}_{t}(v)},\sigma,{m_{t}(v)}} \right)} \right)}}}}}\end{matrix}$ The first product C_(3′) = Π_(v) _(t) _(∈V) _(T)G((−1){circumflex over (^(m))} ^(t) ^((v)), σ, m_(t)(v)) depends only onthe soft outcome {circumflex over (m)} since m is fully determined by m.The condition m _(t)(v) ≠ {circumflex over (m)}_(t)(v) in the secondproduct is equivalent to the presence of a soft flip in the measurementof v at step t, which corresponds to ε(e) = 1 for the soft edge e ={v_(t), v_(t+1)}. Moreover, the$\text{fraction}\frac{\left. {G\left( {\left( {- 1} \right)^{{{\hat{m}}_{t}(v)} + 1},\sigma,{m_{t}(v)}} \right)} \right)}{\left. {G\left( {\left( {- 1} \right)^{{\hat{m}}_{t,}(v)},\sigma,{m_{t}(v)}} \right)} \right)}$coincides with the fraction defining the edge weight of this soft edge ein Eq. (7). This shows Eq. (13) after taking the log. The propositionfollows by taking the log of Eq. (10) and applying Eq. (11), (12), and(13).

While FIG. 4 (above) sets forth general logic that may be employed by adecoding unit to compute a minimum weight solution based on softinformation, the following text sets forth a specific design a softMinimum Weight Perfect Matching decoder along with a proof (Table 2)that this decoder design returns a most likely error configuration.

Soft Decoding of Surface Codes using a Minimum Weight Perfect MatchingDecoder

Given a hard outcome {circumflex over (m)} for T rounds of measurements,we compute the syndrome ŝ=(ŝ₁, . . . , ŝ_(T)) ϵ

defined by ŝ₁(v)={circumflex over (m)}₁(v) andŝ_(t)(v)=({circumflex over (m)}_(t)(t)+{circumflex over(m)}_(t-1)(v))mod2   (14)for all t=2, . . . , T and for all v ϵ V. The syndrome ŝ is a binaryvector that encodes the exact same data as the hard outcome {circumflexover (m)}. What makes the syndrome more convenient is its graphicalmeaning. Consider a subset of edges of G_(T) represented by a vector ε ϵ

. Recall that the

—boundary of ε is defined to be set of vertices ∂ε⊆V_(T) , incident toan odd number of edges of ε.

TABLE 2  Lemma 2.1 Given an error ε ∈  

 for T rounds of measurement (including soft flip), the set of nodes ofG_(T) supporting non-trivial syndrome bits is the boundary ∂ε of theerror.  Proof. By linearity, it suffices to check this for a singlefault. A single flip (hard or soft) on the vertical edge {v_(t),v_(t+1)} leads to a non-trivial syndrome in ŝ_(t)(v) and ŝ_(t+1)(v),that is on the two endpoints of this vertical edge. Similarly, a qubiterror on the horizontal edge {u_(t), v_(t)} results in a non-trivialsyndrome bits ŝ_(t)(u) and ŝ_(t)(v) corresponding to the nodes u_(t) andv_(t).  Given the outcome m observed, we compute the syndrome ŝ. Basedon Lemma 4.1, the goal of the decoder is to identify a set of edges ε ∈ 

 whose boundary is given by the non-trivial syndrome nodes.  We considerthe distance in the graph G_(T) induced by the edge weights w(e). Thelength of a path in G is defined to be the sum of the weights of itsedges. The distance d_(G) _(T) (u, v) between two vertices u and v, isdefined to be the minimum length of a path connecting u and v. A minimumlength path joining u and v is called a geodesic. Given u, v ∈ V_(T),let geo(u, v) ⊆ E_(T) be the set of edges supporting a geodesicconnecting u and v.  Given the soft decoding graph G_(T) and a syndromeŝ, the distance graph K(ŝ) = (V_(K), E_(K)) is defined to be the fullyconnected graph with vertex set V_(K) = {v_(t) ∈ V_(T) | ŝ_(t)(v) = 1}and with edge weight w({u, v}) = d_(G) _(T) (u, v).  Proposition 2.2 Letm be a soft outcome for T rounds of measurement and let ŝ be thecorresponding syndrome. Consider the distance graph K(ŝ) = (V_(K),E_(K)) associated with ŝ. For any subset M ⊆ E_(K), let ε_(M) ∈  

 be the error configuration with support U_({u,v}∈M) geo(u, v) ⊂ E_(T).If M is a minimum weight perfect matching in K(ŝ), then the errorconfiguration ε_(M) is a most likely error configuration conditioned onthe soft outcome m.  Just like Proposition 1, this result remains validfor general classical noise using the soft edge weights defined in Eq.(8). The proof is identical.  Proof. We will prove the proposition bycontradiction. Assume that there exists an error ε′ such that  

 (ε′|m) >  

 (ε_(M)|m).  By Lemma 4.1, the support of ε′ can be partitioned as aunion of edge-disjoint paths connecting the nodes of the support of ŝ.Each path of this set corresponds to an edge in K(ŝ) and the completeset of paths forming ε′ induces a perfect matching M′ in K(ŝ). Moreover,the weight of the matching M′ is the sum of edge weights of ε′. Based onProposition 1, this weight is equal to   w(M′) = Σ_(e∈E) _(T) w(e)ε′(e)= C − log 

 (ε′|m). Using the assumption  

 (ε′|m) >  

 (ε_(M)|m), this yields   w(M′) = C − log 

 (ε′|m) < C − log 

 (ε_(M)|m) = Σ_(e∈E) _(T) w(e)ε_(M)(e) = w(M) which contradicts theminimality of M.  A consequence of Proposition 2.2 is that one canefficiently compute a most likely error conditioned on a softmeasurement outcome using a Minimum Weight Perfect Matching (MWPM)algorithm. Note that the edge weights in the distance graph are non-negative making the application of the MWPM algorithm straightforward.Algorithm 1 (below) describes the decoding strategy.

By example and without limitation, Table 3 below includes exemplarylogic employed by a Soft MWPM decoder:

TABLE 3 Algorithm 1: Soft MWPM Decoder Require: The soft decoding graphG_(T) for T rounds of measurements, each round being represented by asoft outcome vector m. Ensure: A most likely error configuration ε ∈

 conditioned on the soft outcome m.  1. For each soft edge e, computew_(soft)(e) as a function of m using   Eqn.(7).  2. Compute the syndromeŝ from m. Let v₁, . . . , v_(k) be the set of nodes   of G_(T) withnon-trivial syndrome.  3. For i = 1, . . . , k − 1 do:    a. RunDijkstra's algorithm to compute all the distances d_(G) _(T) (v_(i),v_(j))     with j > i.    b. Construct the distance graph K(ŝ) withvertex set V_(K) =     {v₁, . . . , v_(k)} and with edge weightw({v_(i), v_(j)}) = d_(G) _(T) (v_(i), v_(j)).    c. Compute a minimumweight perfect matching M in K(ŝ) using     Edmond's algorithm.    d.For each edge {u, v} ∈ M, compute a geodesic geo(u, v) in the     graphG_(T).    e. Return the error configuration ε supported on the union ofthe     geodesics geo(u, v) for {u, v} ∈ M.

The primary difference between the standard hard MWPM decoder and theSoft MWPM decoder proposed above in Table 3 is that, in the Soft MWPMdecoder, the weights in the soft decoding graph depend on the inputoutcome m whereas they are fixed in the hard case. As a result, thedistances between all pairs of nodes can be precomputed in the standardMWPM but not in the soft MWPM decoder. Similarly, one cannot precomputea geodesic for each pair of nodes in the soft decoding graph.

The Union-Find decoder, which achieves a good approximation of the MWPMdecoder with complexity 0(d³α(d)), can also be adapted to the correctionof soft noise. The complexity of the soft Union-Find decoder depends onthe precision required for the edge weights of the soft decoding graph.

FIG. 6 illustrates operations computed by different components of anexemplary quantum computing system 600 that uses soft information toimprove performance of a decoding unit. The quantum computing system 600includes a syndrome measurement circuit 602 comprising quantum hardware.The syndrome measurement circuit 602 performs multiple rounds ofsyndrome measurements 606 that each measure a soft outcome vector (e.g.,a syndrome array consisting of soft values) consisting of an array ofsoft bit values. At each round of measurement, the syndrome measurementcircuit 602 outputs the measured soft outcome vector to a soft edgeweight computation engine 604. In one implementation, the soft edgeweight computation engine 604 is implemented as software executable by aclassical processor.

The soft edge weight computation engine 604 computes (at operation 607)soft vertical edge weights for each round of measurement. Soft verticaledge weights are to be understood as values corresponding to verticaledges on a decoding graph that corresponds to the physical layout ofqubits of the syndrome measurement circuit 606. In one implementation,the soft vertical edge weights are computed using Eqn. (7) as providedherein (e.g., based exclusively on soft outcome data). In otherimplementation, the soft vertical edge weights are computed based on amix of hard and soft outcome data for the given measurement round.Although the soft edge weight computation engine 604 is shown separatefrom the decoding unit 608, some implementations of the disclosedtechnology may incorporate some or all logic of the soft edge weightcomputation engine 604 within the decoding unit 608.

The decoding unit 608 performs a sequence of operations to identifyfault locations in the decoding graph and—more generally, to identifythe actual corresponding locations of faults within the syndromemeasurement circuit 602. This logic is based, at least in part, on thesoft vertical edge weights that are computed by the soft edge weightcomputation engine 604 following each round of syndrome measurement.

The decoding unit 608 may be any decoding unit that computes orapproximates a minimum weight solution including without limitation, theMinimum Weight Perfect Matching (MWPM) decoder, the Union-Find (UF)decoder, or other machine learning decoders or tensor network decoders.

Following each round of measurement performed by the syndromemeasurement circuit 602, the decoding unit 608 receives inputs including(1) the soft outcome data measured by the syndrome measurement circuit602; and (2) the soft vertical edge weights (e.g., defined as describedabove with respect to FIG. 4 ) that are computed by the soft edge weightcomputation engine 604 based on the soft outcome data for themeasurement round.

Specific operations of the decoding unit 608 may vary slightly insubstance or in order based on the particular design of the decodingunit 608. In the illustrated example, the decoding unit 608 performs amapping operation 610 in which the received soft outcome values aremapped to hard outcome values, such as given by the relationship setforth in Eqn. (3) above. Following this mapping of the soft outcomevalues to hard outcome values, the decoding unit 608 determines (atoperation 612) edge weights of all edges of a decoding graphrepresenting to the qubit configuration of the syndrome measurementcircuit 602. Determining edge weights comprises operations including:

(1) computing or otherwise determining hard edge weights for horizontaledges (e.g., such as using Eqn. (5), above); and

(2) using the received soft vertical edge weights as the vertical edgeweight values. In another implementation, the decoding unit 608determines vertical edge weights based on a combination of hard and softvertical edge weights (e.g., computed via Eqn. (6) and (7) above,respectively).

Based on the edge weights determined by the operation 612, the decodingunit proceeds to build a distance graph at operation 614. In this step,the decoding unit 608 computes a distance (weight) between all possiblepairs of non-trivial syndrome bits (e.g., in a manner the same orsimilar to that discussed with respect to FIG. 5 ).

At a solution selection operation 616, the decoding unit 608 uses thedistance graph constructed at the operation 612 to identify a minimumweight solution. The minimum weight solution is provided to an errorcorrection block 618 that performs operations (at 620) for correctingclassically correcting errors that impacted measurement(s) performed bythe syndrome measurement circuit 602.

FIG. 7 and the following discussion are intended to provide a brief,general description of an exemplary computing environment in which thedisclosed technology may be implemented. Although not required, thedisclosed technology is described in the general context of computerexecutable instructions, such as program modules, being executed by apersonal computer (PC). Generally, program modules include routines,programs, objects, components, data structures, etc., that performparticular tasks or implement particular abstract data types. Moreover,the disclosed technology may be implemented with other computer systemconfigurations, including hand held devices, multiprocessor systems,microprocessor-based or programmable consumer electronics, network PCs,minicomputers, mainframe computers, and the like. The disclosedtechnology may also be practiced in distributed computing environmentswhere tasks are performed by remote processing devices that are linkedthrough a communications network. In a distributed computingenvironment, program modules may be located in both local and remotememory storage devices. Typically, a classical computing environment iscoupled to a quantum computing environment, but a quantum computingenvironment is not shown in FIG. 7 .

With reference to FIG. 7 , an exemplary system for implementing thedisclosed technology includes a general purpose computing device in theform of an exemplary conventional PC 700, including one or moreprocessing units 702, a system memory 704, and a system bus 706 thatcouples various system components including the system memory 704 to theone or more processing units 702. The system bus 706 may be any ofseveral types of bus structures including a memory bus or memorycontroller, a peripheral bus, and a local bus using any of a variety ofbus architectures. The exemplary system memory 704 includes read onlymemory (ROM) 708 and random access memory (RAM) 710. A basicinput/output system (BIOS) 712, containing the basic routines that helpwith the transfer of information between elements within the PC 700, isstored in ROM 708.

In on implementation, the system memory 704 stores decoding logic 711,such as QECCs and logic specifically implemented by various systemdecoders, such as the MWPM decoder or the UF decoder that may determinevertical edge weights of a decoding graph based on soft information, asdescribed herein.

The exemplary PC 700 further includes one or more storage devices 1230such as a hard disk drive for reading from and writing to a hard disk, amagnetic disk drive for reading from or writing to a removable magneticdisk, and an optical disk drive for reading from or writing to aremovable optical disk (such as a CD-ROM or other optical media).-Suchstorage devices can be connected to the system bus 706 by a hard diskdrive interface, a magnetic disk drive interface, and an optical driveinterface, respectively. The drives and their associated computerreadable media provide nonvolatile storage of computer-readableinstructions, data structures, program modules, and other data for thePC 700. Other types of computer-readable media which can store data thatis accessible by a PC, such as magnetic cassettes, flash memory cards,digital video disks, CDs, DVDs, RAMs, ROMs, and the like, may also beused in the exemplary operating environment.

A number of program modules may be stored in the storage devices 1230including an operating system, one or more application programs, otherprogram modules, and program data. Decoding logic can be stored in thestorage devices 730 as well as or in addition to the memory 704. A usermay enter commands and information into the PC 700 through one or moreinput devices 740 such as a keyboard and a pointing device such as amouse. Other input devices may include a digital camera, microphone,joystick, game pad, satellite dish, scanner, or the like. These andother input devices are often connected to the one or more processingunits 702 through a serial port interface that is coupled to the systembus 706, but may be connected by other interfaces such as a parallelport, game port, or universal serial bus (USB). A monitor 746 or othertype of display device is also connected to the system bus 706 via aninterface, such as a video adapter. Other peripheral output devices 745,such as speakers and printers (not shown), may be included.

The PC 700 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer760. In some examples, one or more network or communication connections750 are included. The remote computer 760 may be another PC, a server, arouter, a network PC, or a peer device or other common network node, andtypically includes many or all of the elements described above relativeto the PC 700, although only a memory storage device 762 has beenillustrated in FIG. 7 . The personal computer 700 and/or the remotecomputer 1260 can be connected to a logical a local area network (LAN)and a wide area network (WAN). Such networking environments arecommonplace in offices, enterprise wide computer networks, intranets,and the Internet.

When used in a LAN networking environment, the PC 700 is connected tothe LAN through a network interface. When used in a WAN networkingenvironment, the PC 700 typically includes a modem or other means forestablishing communications over the WAN, such as the Internet. In anetworked environment, program modules depicted relative to the personalcomputer 700, or portions thereof, may be stored in the remote memorystorage device or other locations on the LAN or WAN. The networkconnections shown are exemplary, and other means of establishing acommunications link between the computers may be used.

A system disclosed herein includes a soft information computation engineand a decoding unit. The soft information computation engine computessoft information quantifying an effect of soft noise on multiple roundsof a syndrome measurement output by a quantum measurement circuit. Inthis context, “soft noise” refers to noise that introduces variabilityin repeated measurements of ancilla qubits due to at least one ofimperfections in a readout device and limited measurement time for therepeated measurements. The decoding unit uses the computed softinformation to identify fault locations that collectively explain thesyndrome measurement output.

In another example system of any preceding system, the soft informationcomputing engine is further configured to receive, from the quantummeasurement circuit, multiple rounds of a soft outcome vector. The softoutcome vector contains real number values representing measurements ofa plurality of syndrome bits, where each of the syndrome bits providesinformation about one or more errors affecting a quantum measurement.The soft information computing engine is further configured to generatea decoding graph defining nodes corresponding to the plurality ofsyndrome bits, where the nodes are connected to one another byhorizontal edges and vertical edges. The soft information computingengine computes a set of soft edge weights, each of the soft edgeweights corresponding to one of the vertical edges in the decoding graphand being based on real number measurement values of a syndrome bitcorresponding to endpoints of the vertical edge.

In yet still another example system of any preceding system, thedecoding unit is further configured to determine a minimum weightsolution for the decoding graph based on the computed soft edge weights.

In still another example system of any preceding system, the soft noiseis distinct from quantum noise arising from bit-flips in data qubitsthat are indirectly measured by the ancilla qubits.

In another example system of any preceding system, the decoding unituses the soft edge weights as edge weights for vertical edges of thedecoding graph.

In still another example system of any preceding system, the softinformation computing engine computes the set of soft edge weights basedon a real number measurement value of each of the syndrome bits and alsobased on a hard outcome value for each of the syndrome bits.

In yet still another example system of any preceding system, thedecoding unit implements logic of at least one of a union find (UF)decoder and a minimum weight perfect matching (MWPM) decoder.

An example method disclosed herein provides for computing softinformation that quantifies an effect of soft noise on multiple roundsof a syndrome measurement output by a quantum measurement circuit, wherethe soft noise introduces variability in repeated measurements ofancilla qubits due to at least one of limited measurement time andimperfections in a readout device. The soft noise is distinct fromquantum noise that arises from bit-flips in data qubits that areindirectly measured by the ancilla qubits. The method further providesfor identifying fault locations within the quantum measurement circuitbased on the computed soft information where the identified faultlocations collectively explain the syndrome measurement output.

In still another example method of any preceding method, computing thesoft information further includes receiving, from the quantummeasurement circuit, multiple rounds of a soft outcome vector thatcontains real number values representing measurements of a plurality ofsyndrome bits providing soft information errors affecting a quantummeasurement. The method further provides for generating a decoding graphdefining nodes corresponding to the plurality of syndrome bits, wherethe nodes are connected to one another by horizontal edges and verticaledges. Finally, the method further provides for computing a set of softedge weights that each correspond to one of the vertical edges in thedecoding graph and is based on real number measurement values of asyndrome bit corresponding to endpoints of the vertical edge.

In another example method of any preceding method, identifying faultlocations further comprises determining a minimum weight solution for adecoding graph based on the computed soft information.

In still another example method of any preceding method, the decodinggraph is a three-dimensional graph that includes multiple layers of a 2Dgrid having nodes that correspond to measurement locations of thesyndrome bits within the measurement circuit. The multiple layers of the2D grid are separated by vertical edges each representing a time stepbetween repeated measurements of the nodes within the 2D grid.

In yet still another example method of any preceding method, determiningthe minimum weight solution further comprises using the computed set ofsoft edge weights as edge weights for corresponding vertical edges ofthe decoding graph.

In another example method of any preceding method, determining theminimum weight solution further comprises deriving edge weights forvertical edges of the decoding graph, each of the edge weights beingbased on a soft edge weight of the computed soft edge weights and afurther based on a computed hard edge weight.

In still another example method of any preceding method, determining theminimum weight solution further comprises building a distance graph withat least one of a union find (UF) decoder and a minimum weight perfectmatching (MWPM) decoder.

An example system disclosed herein includes a means for computing softinformation that quantifies an effect of soft noise on multiple roundsof a syndrome measurement output by a quantum measurement circuit, wherethe soft noise introduces variability in repeated measurements ofancilla qubits due to at least one of limited measurement time andimperfections in a readout device. The soft noise is distinct fromquantum noise that arises from bit-flips in data qubits that areindirectly measured by the ancilla qubits. The method further provides ameans for identifying fault locations within the quantum measurementcircuit based on the computed soft information where the identifiedfault locations collectively explain the syndrome measurement output.

An example computer-readable storage media disclosed herein encodescomputer executable instructions for executing a computer process thatprovides for computing soft information quantifying an effect of softnoise on multiple rounds of a syndrome measurement output by a quantummeasurement circuit. The soft noise includes noise that introducesvariability in repeated measurements of ancilla qubits due to at leastone of limited measurement time and imperfections in a readout device.The computer process further provides for identifying fault locationswithin the quantum measurement circuit based on the computed softinformation, where the identified fault locations collectively explainthe syndrome measurement output.

In another example computer process of any preceding computer process,the computer process further comprises receiving, from the quantummeasurement circuit, multiple rounds of a soft outcome vector. The softoutcome vector contains real number values representing measurements ofa plurality of syndrome bits providing soft information about errorsaffecting a quantum measurement. The computer process further providesfor generating a decoding graph defining nodes corresponding to theplurality of syndrome bits, where the nodes are connected to one anotherby horizontal edges and vertical edges, and also for computing a set ofsoft edge weights, where each of the soft edge weights corresponds toone of the vertical edges in the decoding graph and is based on realnumber measurement values of a syndrome bit corresponding to endpointsof the vertical edge.

In another example computer process of any preceding computer process,identifying the fault locations further comprises determining a minimumweight solution for a decoding graph based on the computed softinformation.

In another example computer process of any preceding computer process,determining the minimum weight solution further comprises: using thecomputed set of soft edge weights as edge weights for correspondingvertical edges of the decoding graph.

In yet still another example computer process of any preceding computer,process, determining the minimum weight solution further comprisesderiving edge weights for vertical edges of the decoding graph, whereeach of the edge weights is based on a soft edge weight of the computedsoft edge weights and further based on a computed hard edge weightcomputed.

In yet still another example computer process of any preceding computerprocess, the minimum weight solution further comprises: employing atleast one of a union find (UF) decoder and a minimum weight perfectmatching (MWPM) decoder to build a distance graph.

The above specification, examples, and data provide a completedescription of the structure and use of exemplary implementations. Sincemany implementations can be made without departing from the spirit andscope of the claimed invention, the claims hereinafter appended definethe invention. Furthermore, structural features of the differentexamples may be combined in yet another implementation without departingfrom the recited claims. The above specification, examples, and dataprovide a complete description of the structure and use of exemplaryimplementations. Since many implementations can be made withoutdeparting from the spirit and scope of the claimed invention, the claimshereinafter appended define the invention. Furthermore, structuralfeatures of the different examples may be combined in yet anotherimplementation without departing from the recited claims.

What is claimed is:
 1. A system comprising: a soft informationcomputation engine that: receives multiple rounds of a syndromemeasurement output by a quantum measurement circuit, the syndromemeasurement including a soft value measured for each bit of a pluralityof bits in a quantum circuit, the soft value measured for each of thebits being either of a first range corresponding to a hard outcomeindicative of non-detection of a fault at a corresponding circuitlocation or of a second range corresponding to a hard outcome indicativeof at least one fault detected at the corresponding circuit location;computes soft information based on the soft values for each of the bits,the soft information indicating a confidence in the hard outcomeassociated with the bit and also quantifying an effect of soft noise onthe multiple rounds of a syndrome measurement output by a quantummeasurement circuit, the soft noise introducing variability in repeatedmeasurements of ancilla qubits due to at least one of imperfections in areadout device and limited measurement time for the repeatedmeasurements; and a decoding unit that uses the computed softinformation to identify fault locations that collectively explain thesyndrome measurement output, the soft information biasing the decodingunit toward selecting solution sets including fault locations thatcorrespond to a subset of the bits characterized by low-confidence hardoutcomes in the syndrome measurement.
 2. The system of claim 1, whereinthe soft information computing engine is further configured to: receive,from the quantum measurement circuit, multiple rounds of a soft outcomevector, the soft outcome vector containing real number valuesrepresenting measurements of a plurality of syndrome bits, each of thesyndrome bits providing information about one or more errors affecting aquantum measurement; generate a decoding graph defining nodescorresponding to the plurality of syndrome bits, the nodes beingconnected to one another by horizontal edges and vertical edges; andcompute a set of soft edge weights, each of the soft edge weightscorresponding to one of the vertical edges in the decoding graph andbeing based on a real number measurement value of a syndrome bitcorresponding to endpoints of the vertical edge.
 3. The system of claim2, wherein the decoding unit is further configured to: determine aminimum weight solution for the decoding graph based on the computed setof soft edge weights.
 4. The system of claim 2, wherein the decodingunit uses the soft edge weights as edge weights for vertical edges ofthe decoding graph.
 5. The system of claim 2, wherein the softinformation computing engine computes the set of soft edge weights basedon the real number measurement value of each of the syndrome bits andalso based on a hard outcome value for each of the syndrome bits.
 6. Thesystem of claim 1, wherein the soft noise is distinct from quantum noisearising from bit-flips in data qubits that are indirectly measured bythe ancilla qubits.
 7. The system of claim 1, wherein the decoding unitimplements logic of at least one of a union find (UF) decoder and aminimum weight perfect matching (MWPM) decoder.
 8. A method comprising:receiving multiple rounds of a syndrome measurement output by a quantummeasurement circuit, the syndrome measurement including a soft valuemeasured for each bit of a plurality of bits in a quantum circuit, thesoft value measured for each of the bits being either of a first rangecorresponding to a hard outcome indicative of non-detection of a faultat a corresponding circuit location or of a second range correspondingto a hard outcome indicative of at least one fault detected at thecorresponding circuit location; computing soft information based on thesoft values for each of the bits, the soft information indicating aconfidence in the hard outcome associated with the bit and alsoquantifying an effect of soft noise on the multiple rounds of a syndromemeasurement output by a quantum measurement circuit, the soft noiseintroducing variability in repeated measurements of ancilla qubits dueto at least one of limited measurement time and imperfections in areadout device, the soft noise being distinct from quantum noise arisingfrom bit-flips in data qubits that are indirectly measured by theancilla qubits; and identifying fault locations within the quantummeasurement circuit based on the computed soft information, theidentified fault locations collectively explaining the syndromemeasurement output, the soft information biasing a decoding unit towardselecting solution sets including fault locations that correspond to asubset of the bits characterized by low-confidence hard outcomes in thesyndrome measurement.
 9. The method of claim 8, wherein computing thesoft information further comprises: receiving, from the quantummeasurement circuit, multiple rounds of a soft outcome vector, the softoutcome vector containing real number values representing measurementsof a plurality of syndrome bits providing soft information errorsaffecting a quantum measurement; generating a decoding graph definingnodes corresponding to the plurality of syndrome bits, the nodes beingconnected to one another by horizontal edges and vertical edges; andcomputing a set of soft edge weights, each of the soft edge weightscorresponding to one of the vertical edges in the decoding graph andbeing based on real number measurement values of a syndrome bitcorresponding to endpoints of the vertical edge.
 10. The method of claim8, wherein identifying fault locations further comprises: determining aminimum weight solution for a decoding graph based on the computed softinformation.
 11. The method of claim 10, wherein the decoding graph is athree-dimensional graph that includes multiple layers of a 2D gridhaving nodes that correspond to measurement locations of the syndromebits within the quantum measurement circuit, the multiple layers of the2D grid being separated by vertical edges each representing a time stepbetween repeated measurements of the nodes within the 2D grid.
 12. Themethod of claim 10, wherein determining the minimum weight solutionfurther comprises: using the computed set of soft edge weights as edgeweights for corresponding vertical edges of the decoding graph.
 13. Themethod of claim 10, wherein determining the minimum weight solutionfurther comprises: deriving edge weights for vertical edges of thedecoding graph, each of the edge weights being based on a soft edgeweight of the computed soft edge weights and a further based on acomputed hard edge weight.
 14. The method of claim 10, whereindetermining the minimum weight solution further comprises: building adistance graph with at least one of a union find (UF) decoder and aminimum weight perfect matching (MWPM) decoder.
 15. One or morenon-transitory computer-readable storage media encoding computerexecutable instructions for executing a computer process comprising:receiving multiple rounds of a syndrome measurement output by a quantummeasurement circuit, the syndrome measurement including a soft valuemeasured for each bit of a plurality of bits in a quantum circuit, thesoft value measured for each of the bits being either of a first rangecorresponding to a hard outcome indicative of non-detection of a faultat a corresponding circuit location or of a second range correspondingto a hard outcome indicative of at least one fault detected at thecorresponding circuit location; computing soft information based on thesoft values for each of the bits, the soft information indicating aconfidence in the hard outcome associated with the bit and alsoquantifying an effect of soft noise on the multiple rounds of a syndromemeasurement output by a quantum measurement circuit, the soft noiseintroducing variability in repeated measurements of ancilla qubits dueto at least one of limited measurement time and imperfections in areadout device; and identifying fault locations within the quantummeasurement circuit based on the computed soft information, theidentified fault locations collectively explaining the syndromemeasurement output, the soft information biasing a decoding unit towardselecting solution sets including fault locations that correspond to asubset of the bits characterized by low-confidence hard outcomes in thesyndrome measurement.
 16. The one or more non-transitorycomputer-readable storage media of claim 15, wherein the computerprocess further comprises: receiving, from the quantum measurementcircuit, multiple rounds of a soft outcome vector, the soft outcomevector containing real number values representing measurements of aplurality of syndrome bits providing soft information about errorsaffecting a quantum measurement; generating a decoding graph definingnodes corresponding to the plurality of syndrome bits, the nodes beingconnected to one another by horizontal edges and vertical edges; andcomputing a set of soft edge weights, each of the soft edge weightscorresponding to one of the vertical edges in the decoding graph andbeing based on real number measurement values of a syndrome bitcorresponding to endpoints of the vertical edge.
 17. The one or morenon-transitory computer-readable storage media of claim 15, whereinidentifying the fault locations further comprises: determining a minimumweight solution for a decoding graph based on the computed softinformation.
 18. The one or more non-transitory computer-readablestorage media of claim 17, wherein determining the minimum weightsolution further comprises: using the computed set of soft edge weightsas edge weights for corresponding vertical edges of the decoding graph.19. The one or more non-transitory computer-readable storage media ofclaim 17, wherein determining the minimum weight solution furthercomprises: deriving edge weights for vertical edges of the decodinggraph, each of the edge weights being based on a soft edge weight of thecomputed soft edge weights and further based on a computed hard edgeweight computed.
 20. The one or more non-transitory computer-readablestorage media of claim 17, wherein determining the minimum weightsolution further comprises: employing at least one of a union find (UF)decoder and a minimum weight perfect matching (MWPM) decoder to build adistance graph.